Effective tools for object shape analysis are important and useful in many imaging applications including medical ones. One such popular and widely used tool is the distance transform (DT) (Rosenfeld et al., Pattern Recog. 1:33-61 (1968); Danielsson, Comput. Graphics Image Process. 14:227-248 (1980); Borgefors, Comput. Vision Graphics Image Process. 27:321-345 (1984); Borgefors, Comput. Vision Image Understanding 64:368-376 (1996)) of an object. For a hard (binary) object, DT is a process that assigns a value at each location within the object that is simply the shortest distance between that location and the complement of the object.
However, until very recently this notion of hard DT have not been applicable on fuzzy objects in a meaningful way (Kaufmann, Introduction to the Theory of Fuzzy Subsets, Vol. 1, Academic Press, New York, 1975; Rosenfeld, Inform. Control 40:76-87 (1979); Bezdek and S. K. Pal, Fuzzy Models for Pattern Recognition, IEEE Press, New York, 1992). The notion of DT for fuzzy objects, called fuzzy distance transform (FDT), becomes more important in many imaging applications because it is often necessary to deal with situations with data inaccuracies, graded object compositions, or limited image resolution (on the order of an object's structure size).
The notion of fuzzy distance is formulated by first defining the length of a path on a fuzzy subset and then finding the infimum of the lengths of all paths between two points. For the fuzzy distance between two points x and y, the space n is defined as the “infimum” (Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, Boca Raton, Fla., 1999) of the lengths of all paths between them. The length of a path n in a fuzzy subset of the n-dimensional continuous space n is defined as the integral of fuzzy membership values along π. Generally, there are an infinite number of paths between any two points in a fuzzy subset and it is often not possible to find the shortest path, if it exists. Thus, the fuzzy distance between two points is defined as the infimum of the lengths of all paths between them. It is demonstrated that, unlike in hard convex sets, the shortest path (when it exists) between two points in a fuzzy convex subset is not necessarily a straight-line segment. For any positive number θ≦1, the θ-support of a fuzzy subset is the set of all points in n with membership values greater than or equal to θ. It is shown that, for any fuzzy subset, for any nonzero θ≦1, fuzzy distance is a metric for the interior of its θ-support.
The FDT is thus defined as a process on a fuzzy subset that assigns to a point its fuzzy distance from the complement of the support. The theoretical framework of FDT in continuous space is extended to digital cubic spaces and it is shown that for any fuzzy digital object, fuzzy distance is a metric for the support of the object. In general, FDT is useful, for example, in feature extraction (Fu et al., IEEE Trans. Comput. 25:1336-1346 (1976)), local thickness or scale computation (Pizer et al., Comput. Vision Image Understanding 69:55-71 (1998); Saha et al., Comput. Vision Image Understanding 77:145-174 (2000)), skeletonization (Srihari et al., in Proceedings of International Conference on Cybernetics and Society, Denver, Colo., pp. 44-49 (1979); Tsao et al., Comput. Graphics Image Process. 17:315-331 (1981); Saha et al., Pattern Recog. 30:1939-1955 (1997)), and morphological (Serra, Image Analysis and Mathematical Morphology, Academic Press, San Diego, 1982) and shape-based object analyses (Borgefors, “Applications of distance transformations,” in Aspects of Visual Form Processing (Arcelli et al., eds.), pp. 83-108, World Scientific, Singapore, 1994). In particular, FDT may be useful in fault detection in integrated circuit chips or in computer motherboard circuits, analysis of the dynamics of a hurricane, etc. FDT will be useful in many medical imaging applications, such as computation of local thickness of trabecular bone or vessels, or morphology-based separation of anatomic structures having similar intensities, e.g., artery-vein separation.
Trabecular or cancellous bone—the type of bone that dominates in the vertebrae and at locations near the joints of long bones (metaphysis and epiphysis)—consists of a network of plates and struts. Bone atrophy as it occurs in osteoporosis, leads to either homogeneous or heterogeneous thinning of the trabecular elements. Besides changes in network connectivity (and thus of the topology of the network) the thickness of the trabeculae most critically determines the mechanical competence and thus resistance to fracture of trabecular bone. Accurate measurement of trabecular thickness is, therefore, of significant interest, for example, to assess the effectiveness of anabolic (bone forming) agents of patients with osteoporosis.
The classical approach toward measuring trabecular thickness has been based on histomorphometry of transiliac bone biopsies (Chavassieux et al., in Osteoporosis, 2 (Marcus et al., eds.) New York: Academic Press, pp. 501-509 (2001)). Typically, the perimeter of the trabeculae is measured in stained sections, and thickness is approximated as the bone area divided by one half of the perimeter (Parfitt et al., J Clin. Invest. 72:1396-409 (1983)). The emergence of imaging technologies, such as micro computed tomography (μ-CT) (Ruegsegger et al., Calcified Tissue International 58:24-29 (1996)) enables reconstruction of three-dimensional images calling for more elaborate techniques for measuring structural thickness.
One model-independent approach involves inscribing spheres into the structure (Hildebrand et al., J. Microscopy 185:67-75 (1997)) in such a manner that trabecular thickness at any location is computed as the diameter of the largest inscribed sphere containing that location. The implementation issues are solved using distance transform and the distance ridge, which provides the set of the center points of largest inscribed spheres. This approach is well suited for high-resolution images that can easily be segmented, but it is bound to fail when significant partial volume blurring is present. The latter is the case in images acquired in the limited spatial resolution regime of in vivo μ-MRI and μ-CT that are beginning to supplant bone biopsy-based methods for structural analysis of trabecular bone (see, e.g., Wehrli et al., Topics in Magnetic Resonance Imaging 13:335-356 (2002).
However, the fuzzy nature of these images, caused by partial volume blurring, virtually precludes binarization. Accordingly, until the present invention a long felt need has remained in this art for a method that obviates segmentation and that can effectively deal with images acquired at a voxel size comparable to the typical trabecular bone thickness. It is a goal therefore, to better understand the fuzzy distance transform (in both continuous and digital spaces), to study its properties, to present a dynamic programming-based algorithm to compute FDT for fuzzy digital objects, and to demonstrate practical applications of the FDT methods.